Standardized regression coefficients remove the unit of measurement of predictor and outcome variables. They are sometimes called betas, but I don’t like to use that term because there are too many other, and too many related, concepts that are also called beta.

There are many good reasons to report them:

- They serve as standardized effect size statistics.
- They allow you to compare the relative effects of predictors measured on different scales.
- They make journal editors and committee members happy in fields where they are commonly reported.

If you use a regression procedure in most software, standardized regression coefficients are reported by default. Or at least an easy option.

But there are times you need to use some procedure that won’t compute standardized coefficients for you.

Often it makes more sense to use a general linear model procedure to run regressions. But GLM in SAS and SPSS don’t give standardized coefficients.

Likewise, you won’t get standardized regression coefficients reported after combining results from multiple imputation.

Luckily, there’s a way to get around it.

A standardized coefficient is the same as an unstandardized coefficient between two standardized variables. We often learn to standardize the coefficient itself because that’s the shortcut. But implicitly, it’s the equivalence to the coefficient between standardized variables that gives a standardized coefficient meaning.

So all you have to do to get standardized coefficients is standardize your predictors and your outcome.

How?

### The Steps

Remember all those Z-scores you had to calculate in Intro Stats? It wasn’t the useless exercise you* thought* it was at the time.

Converting a variable to a Z-score is standardizing.

In other words, do these steps for Y, your outcome variable, and every X, your predictors:

1. Calculate the mean and standard deviation.

2. Create a new standardized version of each variable. To get it, create a new variable in which you subtract the mean from the original value, then divide that by the standard error.

3. Use those standardized versions in the regression.

Could this take a while? Yup.

But if that’s what the journal requires you report, just do it.

A nice advantage, is you can apply it, at least partially, even in regression models that can’t usually accommodate standardized regression coefficients.

For example, in a logistic regression it doesn’t make sense to standardize Y because it’s categorical. But you can standardize all your Xs to get rid of their units.

You can then interpret your odds ratios in terms of one standard deviation increases in each X, rather than one-unit increases.

{ 21 comments… read them below or add one }

Hi Karen,

Thanks for the tip!

In fact, one reviewer has requested this.

Is there a citation for this? Can I cite your book?

Tim

Hi Tim,

I don’t think that’s in the book. You can cite this page. I think there are specifi ways to cite web pages, but it depends on your publication manual.

Hi, I need to perform a meta-analysis using the effect sizes of a continuous outcome from different cohort studies. The outcome can be in different scales, so I need to estimate a pooled effect size (SMD). So, from each study I perfom a multivariate linear regression model, and I am thinking to take out the standardize effect size (standardize coefficient) for a specific group variable included in the model and the relative standard error SE. Then I pool the standardize coefficients an the relative SEs across the studies, so the pooled will be again a SMD. Is it correct to use the SE of the standardize coefficient obtained from the regression?

Thanks.

Cinzia

KAREN mam/sir, You greatly explained manual calculation of Standardized Regression Coefficients and I cross checked it. Hats off to you

Greg is right. However, I just compared intercept v. no intercept models in R with small sample size. These were linear models, 2 and 3 predictors for an outcome wit N’s between 20 and 43. Coefficient estimates, R-squared, and F were all very very close between models. Not sure that if this implies always omitting the intercept when using z-score predictors and outcomes, or leaving it in, but just thought I’d give a simulation update.

Hi Karen,

Great explanation, thank you.

I am regularly using log-transformed variables (they are latencies, which usually need log-transformations to account for normality). Would you standardize (or center) the log-transformed or the original variable; and if it’s the original variable, once it is standardized (or centered), would you log-transform it? Thank you!

Hi,

I am working on time series data, i.e. stock market indexes. I extracted the first principal component from 3 major stock market indexes. Before doing the PCA I had to standardize data of all the 3 indexes, which I did. My question is that the values I get for the first principal component , as I understand, will also be standardized ? and secondly when I use the first principal component in regression as independent variable , do I have to standardize the data of my dependent variable, also another stock market index, before I can run the regression .

This is not a strictly correct recipe.

The reason it is incorrect is that if you have N data and you convert them to z-scores, you spend one degree of freedom in computing the mean and another degree of freedom in computing the variance.

So, you end up with N-2 degrees of freedom spread amongst N data points. What that means is that your N data points are not independent of each other. That should be clear if you imagine you have all but one data; you can compute the last one because you know that the z-scores sum to zero.

So, you have N-2 degrees of freedom, but if you follow this recipe, you don’t tell that to the statistical software. It is written to assume that your data are independent, and it therefore assumes you have N degrees of freedom. As a result, it will get the wrong answer.

If your if your model only uses up a small number of the degrees of freedom in the data, then you may be in good shape. For instance, if you have 1000 data and your model uses 200 degrees of freedom, it’s not going to make much difference. Having 800 or 798 degrees of freedom after building your model will change your computed significance levels by only a small amount, and your results should remain (reasonably) valid.

But if you are building a very complex model that eats up most of the degrees of freedom of your data, this can get you bad results. If you start with (for example) 20 data, and your model has 15 free parameters, then, with z-scores you really have only 18 degrees of freedom in your data, rather than 20 before the model. After the model is built, there are only 3 (rather than 5) degrees of freedom left to estimate variances and significance levels. In that case, the difference between 3 d.o.f. and 5 d.o.f. will cause you to substantially over-estimate the significance of your conclusions.

You can reduce the problem by half if you remove the “intercept” term from your model (as it should always be zero, if you build your model on z-scores). But you will still be off by one degree of freedom (because you normalized the standard deviation of the data) and there’s no easy way to work around that.

Thanks so much for your work. It has saved me numerous times. My question is how to get standardized standard errors to report with the standardized coefficients. I need to report standardized direct, indirect and total effects with associated standard errors (effects decomposition). While I can get the standardized effects form Stata, Std. Err. are only generated in relation to the unstandardized coefficients. Is there a way I can standardize an unstandardized standard error? Any help would be greatly appreciated!!

I need go the opposite direction. I have standardized scores but need to construct the corresponding prediction equation using raw variable readings.

Do you know if the coefficients output from excel data toolspack are standardised or unstandardised?

One quick note about logit models. You correctly point out you shouldn’t standardize a dichotomous variable (I would probably argue not to standardize ordinal or categorical variables as well, as standardization implies continuous), and that you can standardize the X variables going into the model. Keep in mind that logit models are actually already standardized. See Williams (2009): http://www3.nd.edu/~rwilliam/oglm/RW_Hetero_Choice.pdf

… in logit and probit models, coefficients are inherently standardized. Rather than standardizing by rescaling all variables to have a variance of one, as in OLS, the standardization is accomplished by scaling the variables and residuals so that the residual variances are either one (as in probit) or π^2/3 (as in logit). If residual variances differ across groups, the standardization will also differ, making comparisons of coefficients across groups inappropriate.

Logit models can be very tricky to interpret when thinking about omitted variable bias (even if they are uncorrelated with your other independent variables) and when comparing across groups or samples.

Hi, should I use standardized variables for linear mixed effect models?

Thanks

Okay… so I just read how to calculate z-scores, which I understand completely. I guess my confusion is when you say “In other words, do these steps for Y, your outcome variable, and every X, your predictors”. What do you mean by outcome variable and predictors? Outcome variable as in the dependent variable? Predictors as in the independent variables/factors/predictors? In this case, I don’t understand how to calculate the z-scores for the predictors unless they’re numeric. In my case, my predictors are discrete.

Hi Delano,

Mathematically, you can still do it with dummy-coded predictor variables. The interpreation doesn’t make much sense, though, and therefore, it’s usually better to just keep those coded 0/1. Standardized coefficients aren’t really meaningful for categorical predictors.

Hi I’m kind of late with this, but this is a great post! I have been wandering how I could determine the relative effects of my predictors. However, I am not understanding the process. I get lost at step 2 when you say subtract the mean from the original value. What original value? Could you show the process using actual numbers? Thanks!!!

Thank you for post. However, if you have used the Multiple Imputation Method, SPSS will not produce the standardised beta weighs but ALSO it wont produce SDs for the pooled data….what is one to do in this situation? many thanks!

Hi Tamlyn, just standardize all Xs and Y BEFORE doing the multiple imputation.

I need to calculate scale scores after I complete the MI (e.g. total anxiety score), but then this becomes a predictor in my regression. This means I can’t standardize the variable prior to running MI, so I am still struggling with how to find the pooled SDs.

Thanks for the above,

I was going to book an hour of consultation for this.

Your site and the workshops are really amazing.

Best wishes

Thanks, Aziz!